Superradiance and scattering of the charged Klein-Gordon field by a step-like electrostatic potential

We develop the scattering theory for the charged Klein-Gordon equation on R-t x R-x, when the electrostatic potential A(x) has different asymptotics a(+/-) as x --> +/-infinity. In this case, the conserved energy is not positive definite (Klein paradox). We construct the spectral representation for the harmonic equation. Since a(+) not equal a(-), the distorted Fourier transform has to be defined on weighted L-2-spaces, and spectral quantities of a new type can appear, that are neither eigenvalues, nor resonances. These so called hyperradiant modes are real singularities of the Green function, and lead to solutions polynomially increasing in time. We investigate the asymptotic behaviours of the solutions as t --> +/-infinity, and we establish the existence of a Scattering operator the symbol of which has a norm strictly larger than 1, for the frequencies in (a(-), a(+)). We apply these results to the DeSitter-Reissner-Nordstrom metric, to justify rigorously the notion of superradiance of charged black-holes. (C) 2004 Elsevier SAS. All rights reserved.